OPTIMIZATION: Point on the line closest to another point

Optimization problems are an application of derivatives in calculus that allow us to find the local and global extrema of a function, including the local and global minima and the local and global maxima.

In order to find the extrema of a function, you'll need to take the derivative of the function, set it equal to zero, and solve it for the independent variable in order to find critical numbers. These critical numbers represent potential critical points, which are the points at which the function changes direction from increasing to decreasing, or vice versa.

Once you've found the critical numbers, you'll need to use the first derivative test to test each interval between the critical numbers to see where the function is increasing and where it's decreasing. If the function is increasing to the left of the critical number, and decreasing to the right of it, the critical number represents a local maximum, and possibly a global maximum. On the other hand, if the function is decreasing to the left of the critical number and increasing to the right of it, the critical number represents a local minimum, and possibly a global minimum.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[DominooseE]

Was having tremendous difficulty on a homework problem and this explained it brilliantly. Was shocked that no derivatives needed to be taken. Thank you so much!

[ADorkNamedSteph]

You explained every detail so clearly and related each piece of information to the next. THANK YOU!

[michaelking]

Just what the doctor ordered! I am 62 and have a minor in math but it has been awhile. I am using your equations for an animation I am coding in Swift.

[jamesbond_]

When you first stated the problem, I figured this was something you were going to solve as a typical minimization problem; i.e. derivatives set equal to zero. HA! I tried going down that path, and eventually came up with the right answer, but your solution in this video is far easier! Thanks for all your videos -- really appreciate the detailed way you describe things!

[bbymig]

Thanks, you just saved me a solid hour.

[johnnolen]

Neato. No calculus required. Being the analysis geek that I am, I optimized the distance formula subject to the constraint that the optimum point had to lie on the given line. I obtained the same result. I also verified that the optimum point resulted in a minimum of the distance formula by performing the 2nd derivative test for a function of two variables.

[joliettraveler]

thank you, thank you, you save me a lot of time and gave a much better explanation than the book.

[Lexyvil]

Thanks for explaining what optimization is! I'm in Cal III and I have to do this in 3D where we need to find a point on a sphere closest/farthest from a given point.

[kagisodipheko]

UNDERSTANDABLE LECTURE...thank you MAM

[zrodger]

Here in 2024, and I'm trying to calculate the distance from my sister's house to the center line of totality of the eclipse (about a half of a mile!) My math skills have greatly eroded since i was younger and this helped a lot!

[humzamuhammad]

Okay but how do you figure this out on your own without help

[picknikbasket]

Nice, I'd forgotten about 90% of this.

[jemcel]

What if I was asked to find a point closest to a certain equation of a parabola?

[awseverythihgcannel]

thanks a lot, this helped greatly in building my autonomous drone!

[prawinraj]

Why can't we use foot of perpendicular formula

[williamgustafssonknutsson]

Does this work with a curve and point in space?

[facialreactions]

WOAHH!!! I found you! I watched your vids from like 4 years ago, back when I was in highschool. Your videos have definitely helped me. But more importantly, you've opened my eyes in believing that there are girls out there that are smart and pretty hehe ;) I just want to say thank you!

[PoliticalJohn]

I love optimization. It was a nice break in the middle of first semester Calculus =p

[blade]

This was really nice and simple, but unfortunately the question I'm trying to answer says to justify my answer with the second derivative. Just gotta find that too, and I'm done. Thanks for the simple step by step, it helps so much.

[_DD_]

Why didn't you solve it in the same way as the Cone problem? With the distance between (3, 2) and the point P.
d^2 = (2-y) ^2 + (3-x)^2
d^2= (2+2x-3)^2 +(3-x)^2
then you can apply a first derivative set it to 0 and get the points, much faster imo :)